login
A077177
Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes.
0
0, 0, 1, 0, 1, 2, 3, 5, 8, 17, 34, 59, 111, 213, 396, 746, 1413, 2690, 5147, 9826, 18885, 36269, 69952, 134949, 260743, 504636, 978311, 1899832, 3692980, 7190329, 13994206, 27279898, 53195986
OFFSET
1,6
COMMENTS
A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.
Equivalently, number of divisors of s=A070826(n) in the range (sqrt(s), sqrt(2s)). More generally, for any positive integer s, the number of primitive Pythagorean triangles with perimeter 2's equals the number of odd unitary divisors of s in the range (sqrt(s), sqrt(2s)). (A divisor d of n is 'unitary' if gcd(d, n/d) = 1.)
REFERENCES
A. S. Anema, "Pythagorean Triangles with Equal Perimeters", Scripta Mathematica, vol. 15 (1949) p. 89.
Albert H. Beiler, "Recreations in the Theory of Numbers", chapter XIV, "The Eternal Triangle", pp. 131, 132.
F. L. Miksa, "Pythagorean Triangles with Equal Perimeters", Mathematics, vol. 24 (1950), p. 52.
FORMULA
a(n) = A070109(A002110(n)) = A078926(A070826(n)).
EXAMPLE
a(5) = 1 since there is exactly one primitive Pythagorean triangle with perimeter 2*3*5*7*11; its edge lengths are (132, 1085, 1093). a(7) = 3; the 3 triangles have edge lengths (70941, 214060, 225509), (96460, 195789, 218261) and (142428, 156485, 211597).
MATHEMATICA
a[n_] := Length[Select[Divisors[s=Times@@Prime/@Range[2, n]], s<#^2<2s&]]
PROG
(PARI) semi_peri(p)= {local(q, r, ct, tot); ct=0; tot=0; pt=0; fordiv(p, q, r=p/q-q; if(r<=q&&r>0, print(q, ", ", r, " [", gcd(q, r), "] "); if(gcd(q, r)==1, ct=ct+1; if(q*r%2==0, pt=pt+1; ); ); tot=tot+1); ); print("semiperimeter:"p, " Total sets:", tot, " Coprime:", ct, " Primitive:", pt); } /* Lists all pairs q, r such that the triangle with edge lengths (q^2-r^2, 2qr, q^2+r^2) has semiperimeter p. */
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Kermit Rose and Randall L Rathbun, Nov 29 2002
EXTENSIONS
Edited by Dean Hickerson, Dec 18 2002
STATUS
approved